Accumulative Pollution, "Clean
Technology," and Policy Design
Cees Withagen
Michael A. Toman
Discussion Paper 98-43
July 1998
1616 P Street, NW
Washington, DC 20036
Telephone 202-328-5000
Fax 202-939-3460
© 1998 Resources for the Future. All rights reserved.
No portion of this paper may be reproduced without
permission of the authors.
Discussion papers are research materials circulated by their
authors for purposes of information and discussion. They
have not undergone formal peer review or the editorial
treatment accorded RFF books and other publications.
Accumulative Pollution, "Clean Technology,"
and Policy Design
Cees Withagen and Michael A. Toman
Abstract
Environmental policymakers must address the adverse effects of a number of
pollutants that accumulate in the environment. Goals for the regulation of these damages
often involve holding long-term emissions below a level deemed to be "dangerous,'' or
outright banning of offending products or processes along with subsidization of more "green"
alternatives. This paper builds upon previous studies by Keeler, Spence, and Zeckhauser
(1971) and Tahvonen and Withagen (1996) in addressing the optimal long-term management
of an accumulative but assimilatable pollutant through policies that restrict more damaging
production processes and thereby induce more benign alternatives. Using a simple general
equilibrium approach, we consider the possibility that the assimilative capacity of the
environment is diminished and eventually exhausted by pollution accumulation. In this case
there is a nonconvexity in the problem that gives rise to multiple potential optima,
complicating the characterization of the optimal path and the determination of decentralized
policies that can support an optimal outcome. In particular, environmental quality may be
preserved or completely degraded in the long term. This makes the question of whether
polluting processes or products should be banned more complicated and more interesting.
We characterize the circumstances under which a banning policy is consistent with an
intertemporally optimizing path, we investigate the sensitivity of optimal solutions to the cost
of a clean backstop technology, and we discuss more generally the design of price-based and
quantity-based policies for supporting an optimal solution.
Key Words: stock externalities, nonconvexities, sustainable development
JEL Classification Numbers: Q20, Q28, D62
ii
Table of Contents
1.
2.
3.
Introduction .................................................................................................................. 1
The Model .................................................................................................................... 2
Properties of Optimal Paths .......................................................................................... 7
3.1 Nondecreasing Pollutant Decay ........................................................................... 9
3.2 Assimilation Capacity that Decreases and Can Disappear .................................... 9
4. Discrete Technology Choice ........................................................................................12
5. Lessons for Policy Design ...........................................................................................15
6. Concluding Remarks ...................................................................................................18
Appendix ............................................................................................................................19
References ..........................................................................................................................23
List of Figures
Figure 2.1
Figure 3.1
Figure 3.2
Figure 3.3
Figure A1
......................................................................................................................... 5
........................................................................................................................10
........................................................................................................................11
........................................................................................................................13
........................................................................................................................20
iii
ACCUMULATIVE POLLUTION, "CLEAN TECHNOLOGY,"
AND POLICY DESIGN
Cees Withagen and Michael A. Toman*
1. INTRODUCTION
Environmental policymakers must address the adverse effects of a number of
pollutants that accumulate in the environment. Examples include toxic substances like
PCBs and heavy metals, radioactive contamination, biological contaminants in water that
require time to break down, water acidification, stratospheric ozone depletion, and
accumulation of greenhouse gases. For these substances, damages to ecological systems
and human interests depend on the concentration of pollution, and thus in turn on the
accumulation of nondegraded emissions, not just the current emissions flow.
Goals for the regulation of these damages often involve holding long-term emissions
to a level below what is believed to be "dangerous.'' This is the approach taken, for
example, in Article 2 of the United Nations Framework Convention on Climate Change.
Sometimes "dangerous" is interpreted to be a loading within the capacity of the environment
to neutralize or otherwise assimilate damaging effects over the long term.
Rather than regulating damaging emissions, policymakers could simply ban
(immediately or after an adjustment period) the production and use of the offending
substances, if there are substitutes. This was the approach taken for ozone-depleting
chlorofluorocarbons. It is also implicit in, for example, policies to "virtually eliminate''
persistent pollutants in the Great Lakes (International Joint Commission (1989); see also
Foran (1991)). Alternative goods or technologies also could be subsidized to induce a
changeover to their use. Clearly there are relationships among these various strategies, e.g.,
high enough taxes on damaging substances will drive them out of the market and induce
substitution to more benign alternatives. Policies which focus on "green investment'' to
bring forth substitutes in a finite period of time are attracting policy interest as possible
ways to limit environmental damage as well as rejuvenating the economic productivity of
capital stocks.
* The authors' respective affiliations are Department of Economics, Free University Amsterdam and Department
of Economics, Tilburg University; and Resources for the Future, Washington DC. Withagen acknowledges
financial support from the Netherlands Organization for Scientific Research; Toman acknowledges financial
support from the US Environmental Protection Agency. Both authors are grateful for helpful conversations with
or comments from Casper van Ewijk, Karen Palmer, Eva Hentschel, Olli Tahvonen, and seminar participants at
Washington State University, University of Chicago, Tilburg University, Istituto Veneto di Scienze Lettere et
Arti, and the ERE annual meetings. The authors also appreciate very useful comments by anonymous referees.
Address correspondence to Toman at RFF, 1616 P Street NW, Washington DC 20036, USA; toman@rff.org.
1
Withagen and Toman
RFF 98-43
There is a substantial academic literature concerned with optimal control of
accumulative pollutants.1 This paper builds upon previous studies by addressing the optimal
long-term management of an accumulative but assimilatable pollutant through policies that
restrict more damaging production processes and thereby induce more benign alternatives. In
this respect we follow the simple general equilibrium approach taken by Keeler, Spence, and
Zeckhauser (1971) in their "Model II." Unlike these authors, however, we consider the
possibility that the assimilative capacity of the environment is diminished and eventually
exhausted by pollution accumulation, following in particular the partial equilibrium analysis
of Tahvonen and Withagen (1996). In this case there is a nonconvexity in the problem that
gives rise to multiple potential optima, complicating the characterization of the optimal path
and the determination of decentralized policies that can support an optimal outcome.2 In
particular, environmental quality may be preserved or completely degraded in the long term.
This makes the question of whether polluting processes or products should be banned more
complicated and more interesting.
In Section 2 of the paper we give the basic structure and assumptions of the model. In
Section 3 we develop our basic results about the characterization of optimal paths for the two
different classes of pollution degradation function. In Section 4 we consider an interesting
special case of the model with discrete technology options. In Section 5 we consider how the
analysis informs the design of practical policies for dealing with accumulative pollutants. In
Section 6 we offer some brief concluding remarks.
2. THE MODEL
We use a fairly simple model of social surplus maximization in which households'
well-being depends on environmental quality as well as consumption. Pollution occurs as a
byproduct of input utilization by competitive firms producing the consumption good. In the
general case input proportions and thus the pollution intensity of output are variable; however,
restricting the use of the polluting input raises the opportunity cost of producing the
consumption good. Environmental quality is related to the concentration or "stock" of
pollution. In the absence of government policy to internalize environmental externality,
1 See, for example, Keeler, Spence, and Zeckhauser (1971); Plourde (1972); d'Arge and Kogiku (1973); Cropper
(1976); Asako (1980); Heal (1982); Becker (1982); Pethig (1991); van der Ploeg and Withagen (1991); Conrad
and Olson (1992); Falk and Mendelsohn (1993); Hoel (1993); Tahvonen and Kuuluvainen (1993); Tahvonen
(1995, 1996); Tahvonen and Withagen (1996); Tahvonen and Salo (1996); Weyant et al. (1996); and Toman and
Palmer (1997). Here we use the term "optimal'' in the conventional sense to refer to intertemporal programs that
maximize the present value of social surplus. A few studies have been concerned with other social criteria, e.g.,
maximum long-term sustainable surplus with a zero rate of time preference (e.g., Forster (1973); see also
Toman, Pezzey and Krautkraemer (1995) for further discussion and references). In this paper we address only
present-value-maximizing programs.
2 Tahvonen and Withagen do not discuss policies for supporting an optimal path, and their formulation does not
incorporate the possibility of production with a pollution-free technology, which is one of the characteristics of
the Keeler et al. framework.
2
Withagen and Toman
RFF 98-43
production of the consumption good takes place with polluting means and both pollution and
environmental damage accumulate.3
We consider several forms of the pollution assimilation function. One has the quantity
of pollution stock decay or assimilation as an increasing function of the stock of pollution.
This is a reasonable specification if one is thinking about the decay or dispersion of chemical
substances in the environment, like radioactive materials or greenhouse gas concentrations in
the environment. The second specification assumes that degradation or assimilation
monotonically decreases as the pollution stock grows until a saturation level is reached, at
which point assimilative capacity is irreversibly lost. This is a reasonable assumption for
pollutants whose degradation depends on biological action which can be poisoned by
pollution accumulation.4 This threshold introduces a nonconvexity, as we discuss below.5
Finally, the most general case allows for an inverted-U shape for the assimilation function,
with capacity initially rising then falling.
To formalize the presentation, we assume the economy has the capacity to produce a
homogeneous consumer commodity (q ) according to a technology that employs labor (l ) and
some other input ( y ) , for example energy from fossil fuels, that causes pollution. By
appropriate scaling we define the pollution flow also by y . The production technology is
described by a function F with the following properties:
A1) F (l , y ) is concave and continuously differentiable for all (l , y ) > 0 with
nonnegative partial derivatives. Moreover F (0, y ) = 0 and Fly ≥ 0 . We treat the cases
F (l ,0) = 0 and F (l ,0) > 0 separately.
3 For simplicity we do not address pollution occurring directly from product consumption or the prospects for
environmental protection through end-use measures such as recycling.
4 Hediger (1991) distinguishes between pollution assimilation and pollution degradation as follows: if a
pollutant is nondegradable then its accumulation monotonically decreases the assimilative capacity of the
environment, whereas a degradable pollutant may have increasing then decreasing assimilative capacity as a
function of the pollution stock (i.e., an inverted U shape). (See also Fiedler (1992) and Pezzey (1996) for a more
detailed characterization of alternative decay functions and their implications.) We assume that our pollutant is
degradable and can be assimilated, but we distinguish between the physical properties of pollution degradation
and the bioeconomic aspects of pollution assimilation. We assume in particular that the degradation rate either
increases or decreases with the pollution stock, while the assimilation rate monotonically decreases with the
pollution stock (this is reflected in our specification of the damage function). We do not directly address
threshold effects in assimilation that would appear as kinks or discontinuities in the damage function (e.g., the
possibility of environmental "catastrophe'' addressed in Cropper (1976)). However, our results can be extended
to the case of irreversible environmental damage, even when the pollutants do chemically decompose. To see
this, let x denote the pollutant stock, let x denote the saturation level of x, and define y by y = x if x ≤ x and
y = x if x > x . If we represent environmental damage as a function of y rather than x, we can capture the idea
that damage can get stuck at a high level of environmental degradation, even with pollutant decay.
5 Nonconvexity issues also are addressed in Tahvonen and Salo (1996). They are formally dual to the problem
of optimal fisheries management with a nonconcave regeneration function (Cropper, Lee and Pannu 1979).
3
Withagen and Toman
RFF 98-43
The stock of pollutants is denoted by x . It accumulates according to the rate of pollution
y but there is also decay of pollution which is described by the function A, depending on the
existing stock. This gives the following differential equation for the stock of pollutants:
x& = y − A(x) , x(0) = x0 > 0 given.
(2.1)
With respect to the decay function we make three alternative sets of assumptions:
A2.i) A : R + → R + is continuously differentiable and concave. Furthermore,
A(0) ≥ 0, A ′( x ) ≥ 0 for all x ≥ 0 and there exists y ≥ 0 such that A( y ) > 0 .
A2.ii) A : R + → R + is continuous, A(0) > 0 , and there exists x > 0 such that
A( x ) = 0 for all x ≥ x . A is continuously differentiable and concave for x ≥ x , with
A′( x ) ≤ 0 . Finally, x 0 < x .
A2.iii) There exists z > 0 such that A satisfies the conditions of A2.i for 0 ≤ x ≤ z
and the conditions of A2.ii for z ≤ x ≤ x .
A typical representation of the decay function is given in Figure 2.1.
Instantaneous social welfare depends on consumption of the consumer commodity,
leisure and the stock of pollution. The instantaneous welfare function is strongly separable in
these entities. The utility attached to the consumer commodity is given by a function
U satisfying
A3. U :R + → R + is continuously differentiable on R + + and strictly concave, with
U ' (0) = ∞ and U '(∞) = 0 .
Second there is a disutility D associated with the stock of pollutants.
A4. D :R + → R + is continuously differentiable and strictly convex, with
D(0) = D'(0) = 0 and D'(∞) = ∞ .
We assume that leisure, which equals l − l where l is the total amount of time
available, enters linearly into the social welfare function. By appropriate scaling we can
attribute a utility weight of unity to leisure.6 We further assume the usual discounted
6 Note that the formulation of the social welfare function can be obtained in another way as well. It is not
necessary to restrict attention to leisure as an argument in the welfare function. It could as well have been
assumed that instantaneous welfare is obtained from a large set of other commodities besides the consumption
commodity considered here along with pollution, and that utility is quasi-linear in other consumption goods.
4
Withagen and Toman
RFF 98-43
Figure 2.1
A(x)
A2.i
x
A(x)
A2.ii
x
A(x)
A2.iii
5
x
Withagen and Toman
RFF 98-43
utilitarian criterion for analyzing a social optimum, with a positive and constant rate of time
preference ρ . The intertemporal social welfare function then can be written as
∞
∫e
− ρt
[U (q ) − (l − l ) − D ( x )]dt
(2.2)
0
This integral has to be maximized with respect to the labor input and the use of the polluting
input/flow of emissions, taking into account the evolution over time of the stock of pollution
(2.1) and the pure state constraint x ≥ 0 . This is a standard optimal control problem, except
for the pure state constraint, which makes the set of necessary conditions for optimality a little
complicated. We return to this and other technical points below.
Finally, we make one other assumption to guarantee an interior solution for labor and
the pollutant stock.
A5. Suppose D( x ) ≡ 0. Then the problem posed above has a solution, denoted by
(l , y ) such that l c < l and y c > A(0).
c
c
The first part of the assumption says that in the absence of pollution damage not all labor is
allocated to production. This necessarily will also hold when pollution is damaging. The
second part of the assumption says that, in the absence of pollution damage, there exists an
optimal level of the flow of pollution which causes a positive stock of pollution eventually.
The current value Hamiltonian of the problem is
H (l , y , x, µ ) = U ( F (l , y )) + (l − l ) − D( x) + µ[ y − A( x )]
Here, µ is the shadow price of the pollution stock. Intuitively it can be seen that this shadow
price of damage should be negative; we discuss this point formally in the Appendix. To
simplify the notation, define τ = − µ . Given a positive pollution stock along an optimal path,
assumptions A1 and A3 (in particular necessity of labor input and infinite marginal utility at
zero consumption) imply that necessary conditions for an optimum are
∂H / ∂l = 0 : U ' ( q) Fl ( y , l ) = 1
(2.3)
∂H / ∂y ≤ 0, y∂H / ∂y = 0: U ' ( q ) F y ( y, l ) ≤ τ, = τ if y > 0
(2.4)
∂H / ∂x = τ& − ρτ : τ& = − D ' ( x ) + [ρ + A' ( x )]τ
(2.5)
The interpretation of the above conditions is straightforward if τ is viewed as a
pollution tax. Condition (2.3) says that the marginal benefits from labor should equal the
6
Withagen and Toman
RFF 98-43
value of leisure. Condition (2.4) requires that the marginal benefits from using the polluting
input should equal the marginal damage, which equals the tax rate, except in a corner solution
with y=0. Equation (2.5) indicates that the co-state is the current value shadow cost of the
pollution stock. To see this define the present value tax τ and integrate to obtain
∞
∞
t
t
τ (t ) = ∫ e − ρs D' ( x ( s))ds − ∫ τ ( s) A'( x ( s))ds
(2.6)
The first term on the right hand side is the present value of future damage costs from a current
increase in the pollution stock. The second term is the present value of a future change in the
state dynamics, valued at the co-state τ . Our τ has the same interpretation except for the
change in focal date from zero to date t.
An important property of the optimal path is given by Lemma 2.1, which is proved in
the Appendix.
Lemma 2.1. Along an optimal path, x(t) is monotonic and converges.
This property of the optimal solution is useful in constructing graphical representations of the
optimal pollution path and tax rate.
The development to this point has ignored the possibility of the constraint x ≥ 0 being
binding. Lemma 2.2, whose proof also is given in the Appendix, provides a sufficient
condition for treating this constraint as nonbinding.
Lemma 2.2. x (t ) > 0 for all t if and only if ρ + A'(0) > 0.
We assume throughout the rest of the paper that the condition ρ + A' (0) > 0 holds, since
according to Lemma 2.2 the problem of long-term environmental management is not very
significant in the opposite case (in that case, there is some t1 such that x(t ) = 0 for all t ≥ t1
along an optimal path).
3. PROPERTIES OF OPTIMAL PATHS
Since the model contains only one state variable it is in principle possible to perform a
phase diagram analysis. We define for x ≥ 0 and ρ + A'( x ) ≠ 0 the function
τ 1 ( x) =
D ' ( x)
ρ + A' ( x)
(3.1)
In view of (2.5), this gives the locus of points where the optimal tax rate is constant. Due to
the concavity of the decay function and the convexity of the damage function, the locus is
upward sloping wherever it is defined. Moreover, under Assumption A4 we have τ 1 (0) = 0 .
7
Withagen and Toman
RFF 98-43
If there exists x ρ > 0 such that ρ + A'( x ρ ) = 0 then this x ρ is an asymptote. Given (2.5), if
τ > τ 1 ( x) for some x < x ρ then τ& < 0 , and conversely (intuitively, with a lower tax than the
critical level τ 1 ( x ) , usage of the dirty input is encouraged, the pollution stock grows, marginal
damages rise, and the tax needed to internalize damages grows).
The set of points for which the stock of pollution is constant is described by (2.3), (2.4),
and y = A(x ) from (2.1). We denote this locus by τ 2 ( x ) . To derive its properties, assume
first that y = A( x ) > 0 . We can solve (2.3) for l as a function of y, which we denote by
l = ψ ( y ). Straightforward calculations (assuming differentiability) yield
U ' ' Fl Fy + U ' Fly
dψ ( y )
=−
dy
U ' ' Fl Fl + U ' Fll
Then some straightforward but extremely messy algebra exploiting the concavity of the
production function F shows that
sgn (dτ 2 / dx) = −sgn ( A′) ,
(3.2)
Thus, the τ 2 locus is increasing if and only if A′ is decreasing (see the Appendix for further
discussion). Finally, from the construction of τ 2 it follows that if τ < τ 2 ( x) for some x then
x& > 0 , and conversely (intuitively, if the tax rate is lower than the critical level then use of the
dirty input is encouraged and the pollution stock grows).
The analysis in the previous paragraph requires modification in two respects to deal
with corner solutions. If Assumptions A2.ii or A2.iii hold for the pollution assimilation
function A, so that A → 0 as x → x , then τ 2 introduced above is well-defined only for x < x .
For x ≥ x there is no assimilation capacity and the pollution stock can be constant only if
y = 0 , i.e., only if "clean" production is possible.
Now suppose that F (l ,0) > 0 , i.e., completely "clean" production is possible. Define
lˆ > 0 and τˆ > 0 by
U ′( F (lˆ,0)) Fl (lˆ,0) = 1, U ′( F (lˆ,0)) Fy (lˆ,0) = τˆ
(3.3)
Then the locus τ 2 ( x) described above is well-defined only for τ 2 ( x) ≤ τˆ . Intuitively, the point
is that for a pollution tax larger than τˆ , y = 0 and clean production is pursued with l = lˆ .
Many possibilities arise with respect to the phase plane and optimal paths, depending
on the various assumptions invoked. In the balance of this section we illustrate some of the
possibilities, drawing upon related analyses in Tahvonen and Withagen (1996).
8
Withagen and Toman
RFF 98-43
3.1 Nondecreasing Pollutant Decay
If A′ ≥ 0 as in Assumption A2.i, then we have the picture shown in Figure 3.1. The
top panel of this picture shows a unique saddle point equilibrium ( x * , τ * ) with the optimal
steady-state tax satisfying τ * < τˆ . This outcome will obtain in particular if F (l ,0) = 0 ,
meaning y is essential and completely clean production is impossible. The bottom panel
shows an intersection of the τ 1 and τ 2 loci where τˆ < τ * . In this case the long-term steadystate features clean production (y=0) and a pollution stock x̂ given by τ 1 ( x ) = τˆ .
3.2 Assimilation Capacity that Decreases and Can Disappear
The analysis is much more complex if Assumption A2.ii holds for the assimilation
function. As already noted, τ 1 → ∞ as x → x ρ where ρ + A′( x ρ ) = 0 . We also have that
τ 2′ > 0 , and Assumption A5 implies that τ 2 (0) > 0 (since if τ = 0 the optimal choice of y is
y c > A(0) ).
One possible set of outcomes is shown in Figure 3.2, in which we assume that clean
production is possible. In this figure, x ρ < x and τ 1 and τ 2 intersect exactly once at a point
( x * , τ * ) with τ * < τˆ . For any initial point x0 < x there is an extremal path, indicated by α in
the diagram, which converges monotonically to ( x * , τ * ) . Given the possibility of clean
production, part of the convergence from above on this path involves a period of clean
production, but as the extremal path nears its limit use of the dirty input on this path resumes.
There is, however, another possibility. In Figure 3.2 we also assume that x < xˆ ,
where τ 1 ( xˆ ) = τˆ , the tax that triggers use of the clean technology. Suppose that the initial
pollution stock satisfies x ≤ x0 < xˆ . With this environmentally unfavorable starting point,
pollution further accumulates and the optimal pollution tax increases monotonically until the
shadow price of pollution is high enough to trigger the elimination of dirty input use and the
stock of pollutants ceases to grow but remains at a level beyond any assimilative capacity.
This path is shown as β in Figure 3.2.
We can now exploit the fact that solutions of the differential equations for ( x&, τ&) along
the extremal path vary continuously with the initial conditions to argue that path β can be
extended somewhat to the left of x to some pollutant stock level ~
x . Behind this
mathematical construct is the economic observation that the disappearance of assimilative
capacity creates a nonconvexity in the social planner's optimization problem. With this
nonconvexity, the first-order conditions cannot be counted on exclusively to determine the
optimum. In Figure 3.2, if the initial environmental state is in the critical range x ≤ x0 < x ,
then both the α path of environmental restoration to the steady-state x * and the β path to the
9
Withagen and Toman
RFF 98-43
Figure 3.1
t
t 1(x)
τ̂
t*
t 2(x)
x̂
x*
x
(a)
t
t 1(x)
t*
τ̂
t 2(x)
x̂
x*
x
(b)
10
Figure 3.2
τ1 (x )
τ
τ1 (x )
α
τ̂
β
τ2 (x )
τ*
x*
xρ
x
x
11
x̂
x
Withagen and Toman
RFF 98-43
polluted steady-state x̂ are possibilities, and only a direct comparison of the two paths
(through an integral test) can indicate which has higher discounted social surplus.7
As noted, the situation pictured in Figure 3.2 is only one of many possibilities.
Another, somewhat more optimistic situation is shown in Figure 3.3, Here we assume that the
clean technology is more affordable that in Figure 3.2, so that the critical tax rate τˆ is lower
and xˆ < x . In this case there is no prospect of a path starting at x0 < x going beyond the limit
of assimilative capacity like the β path in Figure 3.2, since with these initial conditions in
Figure3.3 the optimal pollution tax will be high enough to trigger immediate use of the clean
technology and the associated environmental recovery.8
On the other hand, if clean technology does not exist (y is an essential input), then in
Figure 3.2 we have the possibility of the pollutant stock exhausting assimilative capacity.
Still other possibilities arise if τ 2 > τ 1 for all x, in which case both the pollution stock and tax
increase and are bounded only by the existence of a clean technology (if at all), or if there are
multiple intersections of these two loci (with stable and unstable potential equilibria but still
the potential for a dirty as well as a cleaner steady-state).
Clearly the most general assumption on the assimilation function, A2.iii, combines the
results of the two cases considered above. Since there is no need to develop these details, we
turn instead to a somewhat closer look at the issue of clean technology choice in a special case.
4. DISCRETE TECHNOLOGY CHOICE
In this section we consider a special case where the economy has the capacity to
produce the consumer commodity in exactly two discrete ways. The first way employs a
"clean" technology, while the second way employs a "dirty" but cheaper technology. We
assume that both technologies are linear in labor:
la = ca qa , lb = cb qb , ca > cb
(4.1)
where l , c and q denote labor input, labor requirement and output respectively and a
and b refer to the clean and the dirty technology respectively. Pollution is a joint product of
technology b, and we assume without loss of generality that the pollution flow also can be
represented by qb. Therefore we have
x& = qb − A(x )
(4.2)
7 A similar outcome occurs if there is no root of the equation A′( x ) + ρ = 0 ; in this case it can be shown that, in
effect, x ρ = x and the analysis is virtually identical to that given in the text.
8 The same kind of outcome would occur if the clean technology were so cheap that it obviated the steady state
given by the intersection of the two τ loci, as shown in the bottom panel of Figure 3.1.
12
Figure 3.3
τ1 (x )
τ
α
τ1 (x )
τ̂
τ2 (x )
τ*
x*
xρ
x̂
x
13
x
Withagen and Toman
RFF 98-43
The current value Hamiltonian now is
H (q , q , x, τ ) = U (q + q ) + l − c q − c q − D ( x) − τ [q − A( x )]
a b
a
b
a a
b b
b
(4.3)
The necessary conditions include the dynamic equation (2.5) for the pollution tax, and
U ′(q a + q b ) ≤ c a ( = c a if q a > 0)
(4.4)
U ′( q a + q b ) ≤ c b + τ ( = c b + τ if q b > 0)
(4.5)
Condition (4.4) says that the marginal instantaneous utility of the non-toxic commodity
should not exceed the marginal cost of producing it. This is also the meaning of (4.5), where
marginal cost of the toxic commodity consists of the sum of marginal cost of production and
the marginal environmental cost or pollution tax.
There exists a strong similarity between the model outlined above and the literature on
renewable natural resources. One could look upon x as the reciprocal of a renewable resource,
which is depleted by extracting the commodity q b , but which is subject to a regeneration
process given by A as a function of the stock, say y = 1/x. The resource stock has amenity
value −D(1 / x ) and the rate of extraction yields a profit U (q b ) − c b q b . The other commodity
is produced according to a backstop technology which, by definition, does not need the natural
resource (see also Levhari and Withagen 1992).
Now define τ$ by
τ$ = c a − c b
(4.6)
If the actual tax is larger than τ$ it is too costly to produce the toxic commodity because its
production cost plus tax exceeds the production cost of the clean commodity. Therefore, τ
will not converge to a value larger than τ$ . It might be that τ converges to τ$ , at which tax
rate producers are indifferent as to what production process they will employ. Alternatively,
τ < τ$ eventually so that producers will only produce the dirty commodity.
Analyzing these cases involves phase plane analysis that is essentially identical to that
presented in the previous section. We can define the locus τ = τ 1 ( x) for τ& = 0 , as before, and
the locus τ = τ 2 ( x ) for x& = 0 in this case is given by (4.5) and
τ 2 ( x ) = U ′( A( x )) − c b
(4.7)
provided A( x ) > 0 and τ < τˆ. These loci have the same properties as described above for the
general model, and thus the solutions also are similar – including the challenges of multiple
extremal paths and the potential for dirty and clean steady states when there is nonconvexity
under Assumption A2.ii. In particular, we have use of the clean technology if and only if
14
Withagen and Toman
RFF 98-43
τ > τˆ . Thus, in outcomes corresponding to the top panel of Figure 3.1, the clean technology
is used only if the pollutant stock is high; as the environment recovers, the economy reverts to
the dirty technology. In the bottom half of Figure 3.1, in contrast, an economy with a
relatively clean environment ( x0 < xˆ ) uses the dirty technology, while an economy with a
dirtier environment ( x0 > xˆ ) relies exclusively on the clean technology. The pattern of
technology use along the β path in Figure 3.2 is even more striking, in that the economy
following this path would avoid using the clean technology until environmental degradation
has gone beyond the limit of assimilative capacity, but would then deploy the clean
technology to prevent the now-unavoidable ongoing flow of pollution damages from growing
too large.9
We emphasize these different patterns of technology choice in the discrete technology
model for two reasons. The first is simply to illustrate the range of possibilities and their
dependence on the underlying features of the problem, including both the behavior of the
assimilation function and the relative cost of the clean technology. These differences have
some implications for policy, which we discuss in the next section. The other reason to
emphasize the patterns is a critical one. In not just the special case we consider here but also
in the general case (and in similar models, e.g. by Keeler et al 1971), it is possible to move
between the clean technology and dirtier technologies at will. There are in particular no fixed
set-up costs governing the deployment or re-deployment of the clean technology. Set-up
costs can make the choice of technique "stickier," and incorporating them into this analysis is
a topic of ongoing research.
5. LESSONS FOR POLICY DESIGN
In this section we combine the analysis from previous sections with reasoning about
optimal policy design related to both pollution control and the stimulation of new
technologies. The first question we address is the desirability of banning dirty products or
processes. As noted in the Introduction, this policy seems to be growing in popularity. The
analysis in Sections 3 and 4 suggests that there is at best a limited rationale for a banning
policy in the case that A′ ≥ 0 . This case presents no fundamental environmental
irreversibilities, and a well-designed environmental policy instrument (see below), perhaps
combined with policies to help ameliorate any failures in the R&D market, will induce
9 George Tolley has pointed out to us that our assumptions about the damage function exclude the plausible and
interesting case in which once assimilative capacity is exceeded, damages are unavoidable but eventually
attenuate over time. We could model this in our setup, for example, by modifying the damage function such that
− λ (t − s )
if x = x at some time s, damages from that time onward (for any x ≥ x ) are given by D ( x ) = D ( x )e
and λ is the rate of damage decay. Our results would be largely unchanged by this modification, except that with
this permanent but attenuating damage the prospects for the β path being optimal in Figure 3.2 would be
enhanced, and there would be no human-induced brake on the accumulation of pollution (no reason to deploy the
clean technology to limit growth in x).
15
Withagen and Toman
RFF 98-43
substitution to the clean technology when this is warranted (taking into account sunk costs in
actual practice as well).
Even with environmental irreversibilities the rationale for a pollution banning policy
remains unclear. In Figure 3.2, suppose that the optimal path for the economy is the clean
path α, rather than β. If the environment starts out degraded but not destroyed ( x * < x0 < x ) ,
and pollution control policy creates a high shadow price consistent with path α, then the
economy will in fact find it attractive to switch to the clean technology or product without a
banning policy.10 Banning is even less warranted if the situation is as shown in Figure 3.3.
Banning thus would be useful in our context only if there is concern that market forces
supplemented by appropriate pollution control policies will not lead the market to the right
technology choices over the longer term (or that this process somehow will take "too long" in
relation to some potential environmental catastrophe). And if this is the case, an important
question which lies largely beyond the scope of this paper is whether banning the offending
products or processes is superior to other policies that correct market failures in the
development and diffusion of new technology, such as R&D support and information/
demonstration campaigns.11
The second question we turn to is the sensitivity of optimal outcomes to key
parameters and the implications of these comparative dynamics for policy. The phase
diagrams highlight the sensitivity of outcomes to the relative costliness of the clean
technology (this is seen most starkly in the discrete choice case, as indicated by equation
(4.6), but it is true for the more general case as well). In particular, Figures 3.2 and 3.3 show
that a fall in the relative cost of the clean technology not only is likely to improve the
environment in general but may also help to avoid the conundrum of dirty versus clean longterm futures with declining assimilative capacity. The obvious message here again is that if
there are R&D or other market failures that impede progress in the development or diffusion
of cleaner technology, there may be a high social value in reducing these impediments.
It is also intuitively obvious that the optimal outcomes are sensitive to the discount
rate employed in the social cost-benefit analysis. It can be shown that an increase in ρ rotates
the τ 1 locus in equation (3.1) to the right and increases the asymptote x ρ . This has the effect
of reducing the prospects for occurrence of the clean steady state in Figure 3.2 versus the dirty
steady state, a result consistent for example with Krautkraemer (1985), among others. Aside
from raising broader issues beyond our scope concerning the nature of social decision criteria
for long-term environmental problems (see, e.g., Azar 1998), and concerns for the sources of
high discounting in practice (e.g., poverty and misfunctioning capital markets), the sensitivity
of the outcome to the discount rate in our analysis also underscores the potential importance
10 Indeed, the question in this case becomes the longer-term viability of the clean technology when the
environment improves. But since our model includes neither sunk costs nor learning-by-doing, it is not really
that capable of addressing this question.
11 A banning policy is a blunt instrument in terms of the timing of switchover; but other policies may encounter
the problem of trying to "pick winners."
16
Withagen and Toman
RFF 98-43
of clean technology development: greater availability of such technology may be able to
increase the prospects for a socially preferred clean outcome.
The third point we address concerns the robustness of various pollution policy designs.
In practice, achieving anything like a dynamically optimal pollution tax may be too much to
hope for. The most that may be possible in practice with tax policy is the setting of a
pollution charge that must be maintained for a long period of time. Consider the limiting case
of this hypothesized institutional rigidity: a fixed pollution charge must be set for all time.
Suppose one knows the desired steady state x* < x$ , or one simply has specified a priori some
other long-term target for pollution stock x # < xˆ . If A′ ≥ 0 then a fixed tax can be set in
Figure 3.1 to support either long-term environmental quality goal. For example, in the
discrete technology case one can determine the desired tax rate τ # by U ′(q b# ) = c b + τ # and
A( x # ) = q b# . In particular, if x # = x * this policy will amount to setting the optimal steadystate tax from the start and having inefficient convergence to the steady-state environmental
quality.
In this case, a small variation in whatever long-term tax is set will trigger only a small
variation in the long-term environmental quality that is induced. This is not the case if
A′( x) < 0 for x < x and A( x) = 0 for x ≥ x . To see this, consider the discrete technology case
and suppose the fixed tax rate τ # is set at a high level (though below τ$ ), leading to a low
level of dirty commodity output, q b# , relative to initial neutralization capacity A( x 0 ) . Then x
will be decreasing, and since A ′ < 0 , there will be an ever growing decrease in x in the future
until x approaches 0. Similarly, if the tax τ # is set too low, x will increase to the limit of the
environment's neutralizing capacity and beyond. In short, a constant tax policy in this case
does not yield a stable solution in terms of the allocation of natural absorptive capacity.
An alternative approach for addressing long-term management of an accumulative
pollutant would be an intertemporal emissions trading policy based on the long-term capacity
of the environment to neutralize pollutants. This approach has received significant recent
attention in connection with the regulation of greenhouse gases (see, e.g., Kosobud et al.
1994). In this approach, one starts again with some specification (optimal or otherwise) of a
long-term environmental target x # and an estimate of the associated pollution neutralization
capacity A( x # ) . This long-term allowable pollution flow is allocated annually through the
economy in some fashion through use-or-lose emissions permits that can be traded among
sources.12 In addition, assuming that x0 < x # , the remaining unexploited capacity x # − x0 can
12 This approach would not necessarily require the capacity to monitor actual pollution releases. For example, if
pollution releases are a readily predictable function of outputs or inputs, as is the case for example with fossil
fuel combustion, then allowances could be assigned based on these observable quantities. The monitoring
problem is more challenging if pollution can be contained or recycled.
17
Withagen and Toman
RFF 98-43
be allocated as a bankable stock of single-use emissions permits that emitters can trade or use
as they see fit.13
This approach would provide greater dynamic efficiency in the adjustment toward a
long-term environmental target than a rigid tax, precisely because the permits market would
be able to induce more dynamically efficient adjustments in the shadow price of the pollutant
over time.14 In particular, the approach would provide reasonable long-term market signals
for the development and diffusion of clean technology. Moreover, the quantity-based
approach would be more robust in the case of decreasing assimilative capacity. The
Weitzman (1974) issue of price versus quantity instruments could argue in favor of a taxbased approach with uncertain costs (see Pizer 1997 for a discussion of this in the context of
greenhouse gases), though this depends on the slope of the marginal damage function.
Moreover, issues of time consistency (Kydland and Prescott 1977) arise in attempting to
implement an intertemporal emissions trading program, but the same issues arise in
connection with any dynamic pollution control program that involves a rising shadow price of
emissions. We therefore conclude that especially in the case of declining and uncertain
assimilative capacity, a dynamic pollution trading program warrants serious consideration as a
control strategy.
6. CONCLUDING REMARKS
We have shown in this paper that the introduction of cleaner substitution possibilities
considerably complicates the dynamics of optimal accumulative pollutant control, especially
with limited assimilative capacity. The analysis does not provide support for easy
prescriptions such as a permanent ban on polluting goods when cleaner ones are available, or
a subsidy for the cleaner good designed to achieve the same end. The analysis instead
underscores the virtues of adapting incentive-based economic instruments to the situation
under consideration, in particular the use of dynamic quantity-based policies.
As we have noted already, an important extension of the current analysis involves
addressing optimal investment in the capacity to produce using cleaner methods. A better
understanding of how product substitution interacts with optimal environmental remediation
policies (extending the work of Levhari and Withagen (1992)) also would be useful. Beyond
these points, an obvious but longer-term objective is the treatment of the implications of
ecological and economic uncertainties.
13 In practice the problem is not so simple since the timing of emissions will affect interim pollutant
concentration; this approach does not ensure the achievement of a particular environmental goal by a fixed time.
14 These cost savings may be substantial, as illustrated by the analysis in Wigley, Richels and Edmonds (1996).
18
Withagen and Toman
RFF 98-43
APPENDIX
In this appendix we prove Lemmas 2.1 and 2.2. We also derive some comparative statics
results that are useful in drawing the phase diagrams of section 3.
The current value Hamiltonian of the problem of section 2 reads
H (l , y , x , µ ) = U ( F (l , y )) + (l − l ) − D( x ) + µ[ y − A( x )]
1. µ(t ) ≤ 0 virtually everywhere.
A necessary condition for optimality is that at each instant of time the Hamiltonian is
maximized with respect to y subject to the condition that y ≥ 0 . Given the fact that the
production function is non-decreasing there exists no maximum if the co-state would ever be
positive.
2. Proof of lemma 2.1.
Suppose on the contrary that x displays a bulge as depicted in Figure A1. Here it is assumed
without loss of generality that x starts increasing at time zero and returns to the old level at
time T.
Given the continuity of x there exist 0 < t1 < t 2 < T such that x (t1 ) = x (t 2 ) > 0, x&(t1 ) > 0 and
x& (t 2 ) < 0. In view of the concavity of F and A and, in particular, the strict concavity of U
and D, the path x is the unique optimal trajectory. But, this being so, the optimal path from t 2
on should be the same as from t1 on, which is not the case by construction. This proves the
monotonicity of the optimal stock of pollutants. Convergence of the stock follows from the
fact that it is bounded from below (by zero) and from above, due to the fact that D( ∞) = ∞ .
3. Proof of lemma 2.2.
First we consider the case where A(0) > 0.
Fix some t . Without loss of generality we take it equal to 0 in order to save on notation.
Consider two adjacent intervals of time with length h starting at 0. The optimal stocks of
pollutants at the beginning of the intervals are denoted by x (0), x (h) and x (2h) . Suppose it
is optimal to have x (0) = x (h) = x (2h) = 0. Then the optimal rates of pollution are
y (0) = y (h) = y (2h) = A(0). These are supposed to be constant along the intervals. The
optimal labor input is constant as well and will be suppressed henceforth. Now we consider
an alternative feasible trajectory, denoted by hats, such that y (0) = A(0) + ε , with ε a
positive constant still to be determined. The labor input is unaltered. We take care that
19
Figure A1
x
~
x
t1
t2
t
t3
20
t1 + h
t
Withagen and Toman
RFF 98-43
x (0) = x (0) = x (2h) = x (2h). Hence y (h) = A(0 + hε ) − ε . The difference between total
welfare along the new and the old paths over the two periods equals:
∆W = − h[U ( F ( A(0)) − D(0)] −
1
h[U ( F ( A(0)) − D(0)] +
1 + ρh
h[U ( F ( A(0) + ε )) − D(0)] +
1
h[U ( F ( A(0) + A(0 + hε ) − ε − A(0)) − D(hε )]
1 + ρh
This expression can be rewritten as follows:
∆W = hε
U ( F ( A(0) + ε ) − U ( F ( A(0))
D ( hε ) − D ( 0 )
1
−h
hε
ε
hε
1 + ρh
+ hε
A(0 + hε ) − A(0)
U ( F ( A(0) + A(0 + hε ) − A(0) − ε ) − U ( F ( A(0))
1
− 1]
[h
hε
A(0 + hε ) − A(0) − ε
1 + ρh
U ( F ( A(0 + ε ) − U ( F ( A(0))


+
(1 + ρh)


hε
ε


=
1 + ρh  A(0 + hε ) − A(0) U ( F ( A(0) + A(0 + hε ) − A(0) − ε ) − U ( F ( A(0)) 
h[
]


hε
A(0 + hε ) − A(0) − ε
hε
D(hε ) − D(0)
h
hε
1 + ρh
The expression between brackets converges to hU ' Fy [ ρ + A'(0)] as ε converges to zero.
Therefore if ρ + A' (0) > 0 the optimal path is overtaken, a contradiction.
−
This procedure cannot be used if A(0) = 0 , because then the flow of pollution would have to
be negative in the second period. But an alternative construction is as follows.
Fix some t. Without loss of generality we take it equal to 0. Suppose x (t ) = 0 for all t ≥ 0.
For this to be optimal it is necessary that F (l ,0) > 0. Consider an alternative feasible path,
denoted by upper bars, constructed as follows. Fix a small positive ε . Choose
l (t ) = l (t ), y (t ) = y (t ) + A(ε ) = A(ε ). And let x (t ) be the solution of
x&(t ) = A(ε ) − A( x ), x (0) = 0. Then x < ε and
21
Withagen and Toman
RFF 98-43
U ( F (l , A(ε )) − U ( F (l ,0)
D(ε ) − D(0)
−ε
A(ε )
ε
which expression is positive for ε small enough, contradicting the optimality of x being
zero.
U ( F (l , y ) − D( x ) − U ( F (l ,0) + D(0) ≥ A(ε )
4. Comparative statics results.
If the stock of pollutants is constant it follows from the necessary conditions 2.1 and 2.3
y = A( x )
U ' ( F (l , y )) Fl (l , y ) = 1
that
dy = A' ( x )dx
and
[U ' ' Fy Fl + U ' Fly ]dy + [U ' ' Fl Fl + U ' Fll ]dl = 0
Total differentiation of 2.4
U ' ( F (l , y )) Fy (l , y ) = τ
and some algebraic manipulation yield
dτ 2
A'
=
{U ' U '[ Fyy Fll − Fly ] + U ' U ''[ Fy2 Fll − 2 Fy Fl Fly + Fl 2 Fyy ]}
dx U ' ' Fl Fl + U ' Fl
In view of the assumptions made, in particular the concavity of U and F and
U ' > 0, U ' ' < 0, Fl > 0, Fy > 0, Fly > 0 , it follows that
sgn
dτ 2
= − sgn A' ( x )
dx
22
Withagen and Toman
RFF 98-43
REFERENCES
Asako, K. 1980. "Economic Growth and Environmental Pollution under the Max-Min Principle,"
Journal of Environmental Economics and Management vol. 7, pp. 157-183.
Azar, C. 1998. "Are Optimal CO2 Emissions Really Optimal? -- Four Critical Issues for Economists
in the Greenhouse," Environmental and Resource Economics (Special Issue), forthcoming.
Becker, R. A. 1982. "Intergenerational Equity: The Capital--Environment Trade-Off'," Journal of
Environmental Economics and Management vol. 9, pp. 165-185.
Conrad, J. M. and L. J. Olson. 1992. "The Economics of Stock Pollutant: Aldicarb on Long Island,"
Environmental and Resource Economics vol. 3, pp. 245-258.
Cropper, M. L. 1976. "Regulating Activities with Catastrophic Environmental Effects," Journal of
Environmental Economics and Management vol. 3, pp. 1-15.
Cropper, M., D. Lee and S. Pannu. 1979. "The Optimal Extinction of a Renewable Natural
Resource," Journal of Environmental Economics and Management vol. 6, pp. 341-349.
d'Arge, R. and K. Kogiku. 1973. "Economic Growth and the Environment," Review of Economic
Studies vol. 40, pp. 61-77.
Falk, I. and R. Mendelsohn. 1993. "The Economics of Controlling Stock Pollutants: An Efficient
Strategy for Greenhouse Gases," Journal of Environmental Economics and Management vol. 25,
pp. 76-88.
Fiedler, K. 1992. "Fundamental Principles of Natural Selfpurification Processes in Water Resources,"
draft manuscript (University-GH-Siegen).
Foran, J. 1991. "The Sunset Chemicals Proposal," International Environmental Affairs, pp. 303-308.
Heal, G.H. 1982. "The Use of Common Property Resources," in V. K. Smith and J. V. Krutilla (eds.),
Explorations in Natural Resource Economics (Baltimore, Md.: The Johns Hopkins University
Press for Resources for the Future).
Hediger, W. 1991. "Environmental Pollution and Assimilative Capacity: Biophysical Limits in an
Economic Analysis," in C. Rossi and E. Tiezzi (eds.), Ecological Physical Chemistry,
Proceedings of an International Workshop, 8-12 November 1990, Sienna, Italy (Amsterdam:
North-Holland).
International Joint Commission. 1989. Great Lakes Water Quality Agreement of 1978 -- as Amended
by Protocol signed November 18, 1987 (Ottawa, Ontario).
Keeler, E., M. Spence, and R. Zeckhauser. 1972. "The Optimal Control of Pollution," Journal of
Economic Theory vol. 4, pp. 19-34.
Kosobud, R., T. A. Daly, D. W. South, and K. G. Quinn. 1994. "Tradable Cumulative CO2 Permits
and Global Warming Control," Energy Journal vol. 15, no. 2, pp. 213-32.
Krautkraemer, J. A. 1985. "Optimal Growth, Resource Amenities and the Preservation of Natural
Environments," Review of Economic Studies vol. 52, no. 1, (January), pp. 153-170.
Kydland, F. and E. Prescott. 1977. "Rules rather than Discretion: The Inconsistency of Optimal
Plans," Journal of Political Economy vol. 85, no. 3 (June), pp. 473-491.
23
Withagen and Toman
RFF 98-43
Levhari, D. and C. Withagen. 1992. "Optimal Management of the Growth Potential of Renewable
Resources," Journal of Economics vol. 56, pp. 297-309.
Pethig, R. 1991. "Problems of Irreversibility in the Control of PMPs," in H. Opschoor and D. Pearce
(eds.), Persistent Pollutants: Economics and Policy (Norwall, Mass.: Kluwer Academic
Publishers).
Pezzey, J. 1996. "An Analysis of Scientific and Economic Studies of Pollution Assimilation," draft
manuscript (University of York).
Pizer, W. A. 1997. Prices vs. Quantities Revisited: The Case of Climate Change, Discussion Paper
98-02 (Washington, D.C.: Resources for the Future).
Plourde, C. G. 1972. "A Model of Waste Accumulation and Disposal," Canadian Journal of
Economics vol. 5, pp. 199-225.
Tahvonen, O. 1995. "Dynamics of Pollution Control When Damage is Sensitive to the Rate of
Pollution Accumulation," Environmental and Resource Economics vol. 5, pp. 9-27.
Tahvonen, O. 1996. "Trade with Polluting Nonrenewable Resources," Journal of Environmental
Economics and Management vol. 30, pp. 1-17.
Tahvonen, O. and J. Kuuluvainen. 1993. "Economic Growth, Pollution, and Renewable Resources,"
Journal of Environmental Economics and Management vol. 24, pp. 101-118.
Tahvonen, O. and S. Salo. 1996. "Nonconvexities in Optimal Pollution Accumulation," Journal of
Environmental Economics and Management vol. 31, pp. 160-177.
Tahvonen, O. and C. Withagen. 1996. "Optimality of Irreversible Pollution Accumulation," Journal
of Economic Dynamics and Control vol. 20, pp. 1775-1797.
Toman, M. and K. Palmer. 1997. "How Should an Accumulative Toxic Substance be Banned?"
Environmental and Resource Economics vol. 9, no. 1 (January), pp. 83-102.
Toman, M., J. Pezzey and J. Krautkraemer. 1995. "Neoclassical Economic Growth Theory and
'Sustainability'," in D. Bromley (ed.), Handbook of Environmental Economics (Cambridge:
Blackwell).
van der Ploeg, F. and C. Withagen. 1991. "Pollution Control and the Ramsey Problem,"
Environmental and Resource Economics vol. 1, pp. 215-236.
Weitzman, M. L. 1974. "Prices vs. Quantities," Review of Economic Studies vol. 41, no. 4, pp. 477491.
Weyant, J., et al. 1996. "Integrated Assessment of Climate Change," in J. P. Bruce, H. Lee, and E. F.
Haites (eds.), Climate Change 1995: Economic and Social Dimensions of Climate Change.
Contribution of Working Group III to the Second Assessment Report of the Intergovernmental
Panel on Climate Change (New York: University of Cambridge Press), pp. 367-396.
Wigley, T. M. L., R. Richels and J. A. Edmonds. 1996. "Economic and Environmental Choices in the
Stabilization of Atmospheric CO2 concentrations," Nature vol. 379, pp. 240-243.
24